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SIGNALS
Evolution, Learning & Information
Brian Skyrms
Signals: Evolution, Learning and Information (as of 12/2008) Brian Skyrms
Table of Contents
Introduction 1. Signals 2. Signals in Nature 3. The Flow of Information 4. Evolution 5. Evolution in Lewis Signaling Games 6. Deception 7. Learning 8. Learning in Lewis Signaling Games 9. Generalizing Signaling Games: Synonyms, Bottlenecks and Other Mismatches 10. Inventing New Signals 11. Networks I: Information Processing 12. Complex Signals and Compositionality 13. Networks II: Teamwork 14. Learning to Network
Introduction
“The word is the shadow of the deed.” – Democritus
In a famous essay on meaning, H. Paul Grice distinguished between natural and
non-natural meaning. Natural meaning depends on associations arising from natural
processes. I say that all meaning is natural meaning.1 This is in the tradition of
Democritus, Aristotle, Adam Smith and David Hume, Darwin, Bertrand Russell and
Ludwig Wittgenstein, David Lewis and Ruth Millikan. It is opposed to Platonism,
Cartesianism, and various stripes of geistphilosophy.
Can the meaning of words arise spontaneously, by chance? We begin by
attacking the question (at a high level of abstraction) with a combination of modern tools.
The first is the theory of signaling games, in whose use we follow the philosopher David
Lewis. The second is the mathematical theory of information, where we replace rather ill-
defined questions about the meaning of words with clearer questions about the
information carried by signals. This uses ideas deriving ultimately from the theory of
Claude Shannon, but elaborated by Solomon Kullback in such a way as to provide a
natural definition of the quantity of information in a signal. One of the original
contributions of this book is a natural definition of the informational content in a signal
that generalizes philosopher’s notion of a proposition as a set of possible worlds. The
third consists in the Darwinian idea of evolution by differential reproduction and natural
variation. In particular we use models of replicator dynamics. The fourth consists of
theories of trial and error learning. The evolutionary question is reposed as a learning
question. Can signals spontaneously acquire information through naive learning in
repeated interaction? The story, even at this simplified abstract level, is much richer than
you might expect. At a more concrete level, there is a vast and growing scientific
literature on signaling in and between cells, neurology, animal signaling, and human
signaling, that we cannot hope to address here. But at our abstract, game theoretic level,
there is one basic point that is clear. Democritus was right.
Is this the end of our story? No, it is the beginning. Signaling systems grow. That
means that signaling games themselves evolve. They are not fixed, closed interaction
structures but rather open structures capable of change. We need to study mechanisms
that can account for such change. There are two stages of this process that are addressed
in this book. One is the invention of new signals. The invention of the original signals
needed to get signaling off the ground is a case in point, but there is also the case of
invention to get out of information bottlenecks. This book introduces a new account – a
new mathematical model – of learning with invention. Invention completely alters the
dynamics of learning in signaling situations. The second stage consists in the
juxtaposition of simple signals to produce complex signals. Complex signals are a great
advance in the evolution of signaling systems. Humans are, of course, very good at this.
But, contrary to some claims, neither syntax nor complex signals are the exclusive
preserve of humans. It is best then, not to think of these not as the results of some
evolutionary miracle, but rather as the natural product of some gradual process.
Signaling transmits information, but it does far more than this. To see this we
need to move further than the simple signaling games with one sender and one receiver.
Signals operate in networks of senders and receivers at all levels of life. Information is
transmitted, but it is also processed in various ways. Among other things, that is how we
think -- just signals running around a very complicated signaling network. Very simple
signaling systems should be able to learn to implement very simple information
processing tasks by very simple means, and indeed they can.
Signaling networks also give a richer view of a dual aspect of signals. Signals
inform action, and signaling networks co-ordinate action. Signaling is a key ingredient in
the evolution of teamwork. You can think of the flow of information in a corporation, or
a government, or a publisher. But you can also think of teamwork in animals, in
cooperative hunting, cooperative breeding, cooperative defense against predators, and
cooperative construction of living spaces. These kinds of teamwork are found not only in
mammals, birds and the insect societies, but also more recently in micro-organisms.
Teamwork is found in bacteria (myxococcus), amoeboids (cellular slime molds), and
algae (volvox). These organisms are models of the transition from unicellular organisms
to multicellularity. And the internal workings of multicellular organisms are themselves
marvels of teamwork. The coordination of the parts in each of these cases is effected by
signals. Of course, in any complex organization, information transmission and processing
and coordination of action may not be entirely separate. Rather, they might be thought of
as different aspects of the flow of information.
Signaling may evolve for various purposes in networks with different structures.
We look at only simple structures that can be thought of as building blocks for larger,
more complex networks. But even at the level of such simple network structures, we have
to think of the network topology itself evolving. The last chapter of this book gives a
little introduction to this field, and introduces novel low-rationality payoff-based
dynamics that learns to network just as well as higher-rationality best-response dynamics.
What is the relation of signaling theory to philosophy? It is epistemology, because
it deals with selection, transmission, and processing of information. It is philosophy of
(proto)-language. It addresses cooperation and collective action – issues that usually
reside in social and political philosophy. It does not quite fit into any of these categories,
and gives each a somewhat novel slant. That’s good, because the theory of signaling is
full of fascinating unexplored questions.
1 Grice is pointing to a real distinction, but in my view it is the distinction between
conventional and non-conventional meaning. Conventional meaning is a variety of
natural meaning. Natural dynamic processes – evolution and learning - create
conventions.
Signals: Evolution, Learning and Information Ch. 1 (06/2008) Brian Skyrms
1. SIGNALS
“Two savages, who had never been taught to speak, but had been brought up
remote from the societies of men, would naturally begin to form that language by
which they would endeavor to make their mutual wants intelligible to each other
…”
Adam Smith
Considerations Concerning the First Formation of Languages
What is the origin of signaling systems? Adam Smith suggests that there is
nothing mysterious about it. Two perfectly ordinary people who did not have a signaling
system would naturally invent one. In the first century BC, Vitruvius says much the same
thing:
“In that time of men when utterance of a sound was purely individual,
from daily habits they fixed on articulate words just as they happened
to come; then, from indicating by name things in common use, the result
was in this chance way they began to talk, and thus originated
conversation with one another.”
Vitruvius is echoing the view of the great atomist, Democritus, who lived four centuries
earlier. Democritus held that signals were conventional and that they arose by chance.1
Can it be true? If so, how can it be true?
The leading alternative view was that some signals, at least originally, had their
meaning “by nature” – that is, that there was an innate signaling system2. At the time this
may have seemed like an acceptable explanation, but after Darwin, we must say that it is
no explanation at all. Bare postulation of an evolutionary miracle is no more explanatory
that postulation of a miraculous invention. Either way, some work needs to be done.
Whatever one thinks of human signals, it must be acknowledged that information
is transmitted by signaling systems at all levels of biological organization. Monkeys3,
birds4, bees, and even bacteria5 have signaling systems. Multicellular organisms are only
possible because internal signals coordinate the actions of their constituents. We will
survey some of the signaling systems in nature in chapter 2. Some of these signaling
systems are innate in the strongest sense. Some are not.
We now have not one but two questions: How can interacting individuals
spontaneously learn to signal? How can species spontaneously evolve signaling systems?
I would like to indicate how we can bring contemporary theoretical tools to bear on these
questions.
2
Sender-Receiver
In 1969 David Lewis framed the problem in a clean and simple way by
introducing Sender-Receiver games.6 There are two players, the sender and the receiver.
Nature chooses a state at random and the sender observes the state chosen. The sender
then sends a signal to the receiver, who cannot observe the state directly but does observe
the signal. The receiver then chooses an act, the outcome of which affects them both,
with the payoff depending on the state. Both have pure common interest – they get the
same payoff – and there is exactly one “correct” act for each state. In the correct act-state
combination they both get positive payoff; otherwise payoff is zero. The simplest case is
one where there are the same number of states, acts and signals. This is where we will
begin.
Signals are not endowed with any intrinsic meaning. If they are to acquire
meaning, the players must somehow find their way to information transmission. Lewis
confines his analysis to equilibria of the game, although more generally we would want
to investigate information transmission out of equilibrium as well. When transmission is
perfect, so that the act always matches the state and the payoff is optimal, Lewis calls the
equilibrium a signaling system. It is a virtue of Lewis’s formulation that we do not have
to endow the sender and receiver with a pre-existing mental language in order to define a
signaling system.
3
That is not to say that mental language is precluded. The state that the sender
observes might be “What I want to communicate” and the receiver’s act might be
concluding “Oh, she intended to communicate that.” Accounts framed in terms of mental
language7, or ideas or intentions can fit perfectly well within sender-receiver games. But
the framework also accommodates signaling where no plausible account of mental life is
available.
If we start with a pair of sender and receiver strategies, and switch the messages
around the same way in both, we get the same payoffs. In particular, permutation of
messages takes one signaling-system equilibrium into another. This fundamental
symmetry is what makes Lewis signaling games a model in which the meaning of signals
is purely conventional.8 It also raises in stark form a question that bothered some
philosophers from ancient times onward. There seems to be no sufficient reason why one
signaling system rather than another should evolve. Of course, there may be many
signaling systems in nature which got an initial boost from some sort of natural salience.
But it is worth considering, with Lewis, the worst case scenario in which natural salience
is absent and signaling systems are purely conventional.
Information in Signals
Signals carry information.9 The natural way to measure the information in a
signal is to measure the extent that the use of that particular signal changes
probabilities.10 Accordingly there are two kinds of information in the signals in Lewis
sender-receiver games: information about what state the sender has observed and
4
information about what act the receiver will take. The first kind of information measu
effectiveness of the sender’s use of signals to discriminate states; the second kind
measures the effectiveness of the signal in changing the receiver’s
res
probabilities of
tion.11
has not been successfully transmitted (or perhaps
at misinformation is transmitted.)
ver acts just as he
ocritus said:
The word is the shadow of the act.12
n this
simple observation. In chapter 3, I will develop a unified framework for both
ac
Both kinds of information are maximal in a signaling-system equilibrium. But this
does not uniquely characterize a signaling system. Both kinds of information can also be
maximal in a state in which the players miscoordinate, and the receiver always does an
act that is wrong for the state. Then there is plenty of information of both kinds, but it
seems natural to say that information
th
Transmission of information clearly consists of more that the quantity of
information in the signal. To deal with this example, you might think that we have to
build in mentalistic concept of information – specifying what the sender intended the
signal to mean and what the receiver took it to mean. Within the framework of Lewis
signaling games this is not necessary. Sender and receiver have pure common interest.
Perfect information about the state is transmitted perfectly if the recei
would if he had direct knowledge of the state. As Dem
A general treatment of information in signaling requires a lot more tha
5
informational quantity and informational content of signals. The notion of informational
content will be new, and will allow a resolution of some philosophical puzzles.
Evolution
As a simple explicit model of evolution, we start with the replicator dynamics.13
It has interpretations both for genetic evolution and for cultural evolution. The population
is large, and either differential reproduction or differential imitation lead the population
proportion of strategy A, p(A), to change as:
dp(A)/dt = p(A) [U(A) – U]
where U(A) is the average payoff to strategy A and U is the average payoff in the
population.
Evolutionary dynamics could operate on one population of senders and another of
receivers as in some cases of interspecies communication, or it could operate on a single
population, where individuals sometimes find themselves in the role of sender and
sometimes in the role of receiver.
Consider the two population model for the simplest Lewis signaling game – 2
states, 2 signals, 2 acts. Nature chooses a state by flipping a fair coin. And for further
simplification, suppose the population only has senders who send different signals for
different states and only receivers who perform different acts when they get different
signals. There are then only two sender’s strategies:
S1: State 1 => Signal 1
6
State 2 => Signal 2 S2: State 1 => Signal 2 State 2 => Signal 1 and only two receiver’s strategies:
R1: Signal 1 => Act 1 Signal 2 => Act 2 R2: Signal 1 => Act 2 Signal 2 => Act 1
The pairs <S1,R1> and <S2,R2> are the signaling system equilibria. (We will consider
varying the numbers of states, signals and acts, and the probabilities of the states, and the
payoffs in subsequent chapters.)
The population dynamics lives on a square, with p(S2), the proportion of senders
playing strategy S2, on the y axis and p(R2), the proportion of receivers playing strategy
R2, on the x axis. It looks like this:
(fig 1 here)
There are 5 dynamic equilibria -the 4 corners and one in the center of the square - but
three of them are dynamically unstable. The 2 signaling systems are the only stable
equilibria, and evolution carries almost every state of the combination of populations to
either one signaling system or another.
Consider a one-population model where the agent’s contingency plans, if sender...
and if receiver ..., correspond to the four corners of the model we just considered. The
dynamics lives on a tetrahedron. It looks like this:
7
(figure 2 here)
The vertices are dynamic equilibria, and in addition there is a line of equilibria running
through the center of the tetrahedron. But again, all the equilibria are unstable except for
the signaling systems. All states to one side of a plane cutting through the tetrahedron are
carried to one signaling system; all to the other side to the other signaling system. Almost
every possible state of the population is carried to a signaling system. More complex
cases are discussed in Chapters 4 and 5.
We see in these simple cases how a perfectly symmetric model can be expected to
yield an asymmetric outcome. In our two examples, the principle of sufficient reason is
defeated by symmetry breaking in the evolutionary dynamics. The population moves to a
signaling system as if - one might say - guided by an unseen hand.
Learning Strategies
As a simple explicit model of unsophisticated learning, we start with
reinforcement according to Richard Herrnstein’s matching law – the probability of
choosing an action is proportional to its accumulated rewards.14 We start with some
initial weights, perhaps equal, assigned to each action. An act is chosen with probability
proportional to its weight. The payoff gained is added to the weight for the act that was
8
chosen, and the process repeats. As the weights build up, the process slows down in
accordance with what psychologists call the law of practice.
Consider repeated interactions between two individuals, one sender and one
receiver, who learn strategies by this kind of reinforcement. This set-up resembles the
two-population evolutionary model, except that the process is not deterministic, but
chancy. For a nice tractable example consider the 2-state, 2-signal, 2-act signaling game
of the last section. Computer simulations show agents always learning to signal, and
learning is reasonably fast.
Learning Actions
We helped the emergence of signaling in the foregoing model by letting
reinforcement work on complete strategies in the signaling game –on functions from
input to output. Essentially, the modeler has done some of the work for the learners. I
take this as contrary to the spirit of Democritus, according to which the learners should
not have to conceive of the problem strategically. Let us reconceptualize the problem by
having reinforcement work on single actions and see if we still get the same result.
To implement this for the simplest Lewis signaling game, the sender has separate
reinforcements for each state. You can think of it as an urn for state 1, with red balls for
signal 1 and black balls for signal 2; and another such urn for state 2. The receiver also
has two urns, one for each signal received, and each containing balls for the two acts.
9
Nature flips a fair coin to choose the state. The sender observes the state and draws a ball
from the corresponding urn to choose a signal. The receiver observes the signal and
draws a ball from the corresponding urn to choose an act. The act is either successful, in
being the act that pays off in that state, or not. Reinforcement for a successful act is like
adding a ball of the color drawn to the sender and receiver urn just sampled. The
individuals are being reinforced for “what to do on this sort of occasion”. We can then
ask what happens when these occasions fit together to form a signaling game.
This model appears to be more challenging than the one in the previous section.
There are now four interacting reinforcement processes instead of two. Equilibria where
the sender ignores the state and the receiver ignores the signal are no longer ruled out by
appeal to the agents’ intelligence and good intentions. Nevertheless, there is now an
analytic proof15 that reinforcement learning converges to a signaling system with
probability one. The robustness of this result over a range of learning rules is discussed in
Chapters 6 and 7.
States, Acts and Signals
In the simplest Lewis signaling games, the number of states, acts and signals are
assumed to be the same. Why should this be so? What if there is a mismatch? There may
be extra signals, or too few signals, or not enough acts. All of these possibilities raise
questions that are interesting both philosophically and mathematically.
10
Suppose there are too many signals. Do synonyms persist, or do some signals fall
out of use until only the number required to identify the states remain in use? Suppose
there are too few signals. Then there is, of necessity, an information bottleneck. Does
efficient signaling evolve; do the players learn to do as well as possible? Suppose there
are lots of states, but not many acts. How do the acts affect how the signaling system
partitions the states?
If we have 2 states, 2 acts and 3 signals, we could imagine that the third signal
gets in the way of efficient signaling, or that one signal falls out of use and one ends up
with essentially a 2 signal system, or that one signal comes to stand for one state and the
other two persist as synonyms for the other state. Simulations of the learning process of
the last section always produce efficient signaling, often with the persistence of
synonyms. Learning is about as fast as in the case where there are only 2 signals.
If we have 3 states, 3 acts and only 2 signals, there is an information bottleneck.
The best that the players could do is to get it right 2/3 of the time. This could be managed
in various ways. The sender might use signals deterministically to partition the states –
for example, send signal 1 in state 1 and signal 2 otherwise. An optimal receiver’s
strategy in reply would be to do act 1 when receiving signal 1, and to randomize between
acts 2 and 3 with any probability. This identifies a whole line of equilibria, corresponding
to the randomizing probability. Alternatively, the receiver could be deterministic – for
example, doing act 1 for signal 1 and act 2 for signal 2. If so, an optimal senders strategy
to pair with this would always do sending signal 1 in state 1 and signal 2 in state 2, but
11
randomizing in state 3. This identifies another line of efficient equilibria.16 There are, of
course, also lots of inefficient equilibria. Simulations always deliver efficient equilibria.
They are always of the first kind, not the second. That is to say the signaling system
always partitions the states. Learning is still fast.
If we have 3 states, but only 2 signals and 2 acts, we can have act 1 right for state
1, and act 2 right for state 3, and then vary the payoffs for state 2:
Payoffs State 1 State 2 State 3Act 1 1 1-e 0 Act 2 0 e 1
If e>.5 it is best to have one signal (which elicits act 1) sent in both state 1 and state 2;
and the other signal (which elicits act 2) sent in state 3. If e> .5 an efficient equilibrium
lumps states 2 and 3 together. The optimal payoff possible depends on e: 2/3 for e = .5
and 1 for e = 0 or e = 1. For the whole range of values, optimal signaling emerges. These
generalized signaling games are discussed in Chapter 8. The signaling game itself may
not be fixed. The game structure itself may evolve. A model of signaling with invention
of new signals is introduced in Chapter 9. The combination of simple signals to form
complex signals is discussed in Chapter 11.
Signaling Networks
Signaling is not restricted to the simple 1-sender, 1-receiver case discussed so far.
Alarm calls usually involve one sender and many receivers, perhaps with some of the
12
receivers being eavesdroppers from other species. Quorum signaling in bacteria has many
individuals playing the role of both sender and receiver. The brain continually receives
and dispatches multiple signals, as do many of its constituents. Most natural signaling
occurs in networks. A signaling network can be thought of as a directed graph, with an
edge directed from node A to node B signifying that A sends signals to B. All our
examples so far have been instantiations of the simplest possible case; one sender sends
signals to one receiver.
•→•
There are other simple topologies that are of interest. One that I discussed
elsewhere17 involved multiple senders and one receiver. I imagined two senders who
observed different partitions of the possible states.
•→•←•
In the context of alarm calls, if one sender observes a snake or leopard is present, and
another observes that there is no snake, a receiving monkey might be well-advised to take
the action appropriate to evade a leopard. Multiple senders who transmit different
information leave the receiver with a problem of logical inference. It is not simply the
problem of drawing a correct inference, but rather the problem of drawing the correct
inference relevant to his decision problem. For instance, suppose sender 1 observes the
truth value of p and then sends signal A or signal B, and sender 2 observes the truth value
of q and sends C or D. Maximum specificity is required where the receiver has 4 acts,
one right for each combination of truth values. But a different decision problem might
13
require the receiver to compute the truth value (p exclusive or q) and of one act if true
and another if false.
Senders may observe different aspects of nature by chance, but they might also be
able to choose what they observe. Nature may present receivers with different decision
problems. Thus, a receiver might be in a situation where he would like to ask a sender to
make the right observation. This calls for a dialogue, where information flows in both
directions.
•↔•
Nature flips a coin and presents player 2 with one or another decision problem. Player 2
sends one of two signals to player 1. Player 1 selects one of two partitions of the state of
nature to observe. Nature flips a coin and presents player one with the true state. Player
one sends one of two signals to player 2. Player 2 chooses one of two acts. Here a
question and answer signaling system can guarantee that player 2 always does the right
thing.
A sender may distribute information to several receivers.
•←•→•
One instance is the case of eavesdropping, where a third individual listens in to a two-
person sender-receiver game, with the act of the third person having payoff consequences
for himself, but not for the other two.18 In a somewhat more demanding setup, the sender
sends separate signals to multiple receivers who then have to perform complementary
14
acts for everyone to get paid. For instance, each receiver must choose one of two acts,
and the sender observes one of four states of nature and sends one of two signals to each
receiver. Each combination of acts pays off in exactly one state.
Signalers may form chains, where information is passed along.
•→•→•
In one scenario, the first individual observes the state and signals the state, and the
second observes the signal and signals the third, which must perform the right act to
ensure a common payoff. There is no requirement that the second individual sends the
same signal that she receives. She might function as a translator from one signaling
system to another.
When we extend the basic Lewis game to each of these networks, computer
simulations show reinforcement learning converging to signaling systems – although a
full mathematical analysis of these cases remains to be done. It is remarkable that such an
unsophisticated form of learning can arrive at optimal solutions to these various
problems. Simple signaling networks are discussed as a locus of information processing
in Chapter 10 and as a component of teamwork in Chapter 13.
These networks are the simplest examples of large classes on phenomena of
general interest. They also can be thought of as modules, which appear as constituents of
more complex and interesting networks that process and transmit information. It is
possible for modules to be learned in simple signaling interactions, and then assembled
15
into complex networks by either reinforcement or some more sophisticated form of
learning. The analogous process operates in evolution. The dynamics of formation of a
simple signaling network is discussed in Chapter 14.
Conclusion
How do these results generalize? This is not so much as single question as an
invitation to explore an emerging field. Even the simplest extensions of the models I have
shown here are full of surprising and interesting phenomena. We have seen the
importance of focusing on adaptive dynamics. The dynamics can be varied. On the
evolutionary side, we can add mutation to differential reproduction. In addition, we might
move from the large population, deterministic model of the replicator dynamics to a small
population stochastic model. The mathematical structure of one natural stochastic model
of differential reproduction is remarkably similar to our model of reinforcement
learning.19 On the learning side, we should also consider more sophisticated types of
learning. From considering evolution in a fixed signaling game we might move to
evolution of the game structure itself. We should explore both signaling on various kinds
of networks, but also the dynamics of formation of signaling networks. The rest of this
book is an introduction to these topics.
We started with a fundamental question. Suppose we start without pre-existing
meaning. Is it possible that, under favorable conditions, unsophisticated learning
dynamics can spontaneously generate meaningful signaling? The answer is affirmative.
16
The parallel question for evolution turns out to be not so different, and is answered in the
same way. The adaptive dynamics achieves meaning by breaking symmetry. Democritus
was right. It remains to explore all the ways in which he was right.
17
figure 1: Replicator Dynamics, Two Populations
18
figure 2: Replicator Dynamics, One Population
19
20
Notes:
1 Another echo is to be found in Diodorus of Sicily: “The sounds they made had no sense and were confused; but gradually
they articulated their expressions, and by establishing symbols among
themselves for every sort of object they came to express themselves on
all matters in a way intelligible to one another. Such groups came into
existence throughout the inhabited world, and not all men had the same
language, since each group organized their expressions as chance had it.”
Translation from Barnes 2001, 221. See also Verlinski 2005 and Barnes 2001, 223. Proclus says:
“Both Pythagoras and Epicurus were of Cratylus’ opinion. Democritus and
Aristotle were of Hermongenes” (5,25-26).
and:
“Democritus who said that names are conventional formulated this principle in
four dialectical proofs…. Therefore names are arbitrary, not natural.” (6,20-7,1)
Translation from Duvick 2007.
2 I am, of necessity, drastically oversimplifying the ancient debate here. See van den Berg
2008.
3 Cheney and Seyfarth 1990.
21
4 See Charrier and Sturdy 2005 for an avian signaling systems with syntactical rules,
and Marler 1999 for shadings of “innateness” in sparrow songs.
5 See the review article of Taga and Bassler 2003.
6 Russell 1921 is a precursor to Lewis. In an important paper, Crawford and Sobel 1982
analyze a model which generalizes signaling games in a different direction than that
pursued here.
7 Such as Hurford 1989 and Komarova, Niyogi and Nowak 2001.
8 Some signaling interactions may not have this strong symmetry and then signals may
not be perfectly conventional. There may be some natural salience for a particular
signaling system. Here we are addressing the worst case for the spontaneous emergence
of signaling.
9 I follow Dretske 1981 in taking the transmission of information as one of the
fundamental issues of epistemology.
10 This can be measured in a principled way using the discrimination information of
Kullback and Leibler 1951, Kullback 1959. We will look at this more closely in Chapter
3.
22
11 Corresponding to these two types of information, we can talk about two types of
content of a signal. See Russell 1921, Millikan 1984, Harms 2004.
12 Barnes 1982, 468.
13 For a canonical reference, see Hofbauer and Sigmund 1998.
14 First proposed in Herrnstein 1970 as a quantification of Thorndike’s law of effect,
later used by Roth and Erev 1995 to model experimental human data on learning in
games, by Othmer and Stevens 1997 to model chemotaxis in social bacteria, and by
Skyrms and Pemantle 2000 to model social network formation.
15 Argiento, Pemantle, Skyrms and Volkov 2008.
16 Notice that these 2 lines share a point. If we consider all the lines of efficient equilibria,
we have a cycle.
17 Skyrms 2000, 2004.
18 There are also more complicated forms of eavesdropping, where the third party’s
actions have consequences for the signalers and there is conflict of interest. For a
fascinating instance, where plants eavesdrop on bacteria, see Bauer and Mathesius 2004.
23
19 Schreiber 2001, Benaim, Schreiber and Tarres 2004.
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